Question 1: Find the average rate of change of the following functions over the indicated intervals:

Solution

a. y = x2 + 4, from x = 2 to x = 3

Suppose f(x) = x2 + 4, then f(x+Δx) = (x+Δx)2 + 4 The average rate of change is given by Put the values, Simplifying, OR

Now the given interval is from x = 2 to x = 3. This means
Δx = 3 – 2 = 1
Put this value in the above equation
Δy/Δx = 2(2) + 1 = 4 + 1 = 5 Answer.

### b. y = x2 + 1/3 x<, from x = -3 to x = 3

Let y = f(x) = x2 + 1/3 x

The rate of change is given by

Put the values then

Now multiplying the numerator and denominator by 3 and cancelling the concerned terms

Cancelling Δx,

Since the given interval is from -3 to +3, i-e, 6 units in the positive direction, therefore, Δx = 6. Put values

### c. S = 2t3 – 5t +7

Let S = f(t) = 2t3 – 5t +7 then
f(t + Δt) = 2(t + Δt)3 – 5(t + Δt) + 7

Therefore, rate of change is

Cancelling similar terms with opposite signs,

Taking Δt common in the numerator and cancelling it with the denominator, we get,

Since the given interval is from t = 1 to t = 3, therefore, Δt = 3 – 1 = 2. Therefore,

### d. from t = 8 to t = 8.5

Let

Then

And the rate of change of h is

Now from the given data, t = 8 and Δt = 0.5, therefore,

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